Sounds like a freewheel to me.
That last cog should be self tightening. Something has happened to it and allowed at least one other to be free.
If you can't get it to thread back together and tighten properly, you probably need a new freewheel.
6sp freewheels are available. The old one will be tough to get off. You'll need the tool you posted, a good wrench, and maybe a piece of pipe over the wrench to get enough leverage.
Instructions here. The new one should just thread on the (greased) hub threads.